{\it general, non-threshold} ${\cal Q}^2$ adversary structure. However, their protocol requires {\it at least three} rounds
and performs {\it exponential} (exponential in the size of the adversary structure)
computation and communication. They have left it as an open problem
to design efficient protocol for almost perfectly secure
message transmission, tolerating ${\cal Q}^2$ adversary structure.

In this paper, we show the first {\it single}
round almost perfectly secure message transmission protocol
tolerating ${\cal Q}^2$ adversary structure. The

computation and communication complexities of the protocol are both {\it

When we restrict our general protocol to a {\it threshold adversary},
we obtain a single round, {\it communication optimal} almost secure message transmission protocol
tolerating threshold adversary, which is
much more {\it computationally efficient} and {\it
relatively simpler} than the previous single round, communication optimal protocol of
Srinathan et al. (PODC '08).